Bayesian Analysis

A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions

Fabrizio Ruggeri, Zaid Sawlan, Marco Scavino, and Raul Tempone

Full-text: Open access


In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.

Article information

Bayesian Anal., Volume 12, Number 2 (2017), 407-433.

First available in Project Euclid: 12 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

linear parabolic PDEs noisy boundary parameters Bayesian inference heat equation thermal diffusivity

Creative Commons Attribution 4.0 International License.


Ruggeri, Fabrizio; Sawlan, Zaid; Scavino, Marco; Tempone, Raul. A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions. Bayesian Anal. 12 (2017), no. 2, 407--433. doi:10.1214/16-BA1007.

Export citation


  • Bär, M., Hegger, R., and Kantz, H. (1999). “Fitting partial differential equations to space-time dynamics.” Physical Review E, 59(1): 337–342.
  • Charrier, J. (2012). “Strong and weak error estimates for elliptic partial differential equations with random coefficients.” SIAM Journal on Numerical Analysis, 50(1): 216–246.
  • Conrad, P. R., Girolami, M., Särkkä, S., Stuart, A., and Zygalakis, K. (2015). “Probability measures for numerical solutions of differential equations.” arXiv:1506.04592.
  • dos Santos, W. N., Mummery, P., and Wallwork, A. (2005). “Thermal diffusivity of polymers by the laser flash technique.” Polymer Testing, 24(5): 628–634.
  • Evans, L. C. (1998). Partial Differential Equations. American Mathematical Society.
  • Fudym, O., Orlande, H., Bamford, M., and Batsale, J. (2008). “Bayesian approach for thermal diffusivity mapping from infrared images with spatially random heat pulse heating.” Journal of Physics: Conference Series, 135(1): 012042.
  • Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper).” Bayesian Analysis, 1(3): 515–534.
  • Ghosh, J. K., Delampady, M., and Samanta, T. (2006). An Introduction to Bayesian Analysis. Springer.
  • Johnson, C. (1987). Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press.
  • Kaipio, J. P. and Fox, C. (2011). “The Bayesian framework for inverse problems in heat transfer.” Heat Transfer Engineering, 32(9): 718–753.
  • Kullback, S. (1987). “The Kullback–Leibler distance.” The American Statistician. Letters to the Editor, 41(4): 340–341.
  • Kullback, S. and Leibler, R. (1951). “On information and sufficiency.” The Annals of Mathematical Statistics, 22(1): 79–86.
  • Lanzarone, E., Pasquali, S., Mussi, V., and Ruggeri, F. (2014). “Bayesian estimation of thermal conductivity and temperature profile in a homogeneous mass.” Numerical Heat Transfer, Part B: Fundamentals, 66(5): 397–421.
  • Long, Q., Scavino, M., Tempone, R., and Wang, S. (2013). “Fast estimation of expected information gains for Bayesian experimental designs based on Laplace approximations.” Computer Methods in Applied Mechanics and Engineering, 259(1): 24–39.
  • Massard, H., Fudym, O., Orlande, H., and Batsale, J. (2010). “Nodal predictive error model and Bayesian approach for thermal diffusivity and heat source mapping.” Comptes Rendus Mécanique, 338(7–8): 434–449.
  • Müller, T. G. and Timmer, J. (2002). “Fitting parameters in partial differential equations from partially observed noisy data.” Physica D: Nonlinear Phenomena, 171(1): 1–7.
  • Polson, N. G. and Scott, J. G. (2012). “On the half-Cauchy prior for a global scale parameter.” Bayesian Analysis, 7(4): 887–902.
  • Ramsay, J. O., Hooker, G., Campbell, D., and Cao, J. (2007). “Parameter estimation for differential equations: a generalized smoothing approach.” Journal of the Royal Statistical Society, Series B, 69(5): 741–796.
  • Rasmussen, C. E. and Williams, C. K. (2006). Gaussian Processes for Machine Learning. the MIT Press.
  • Ruggeri, F., Sawlan, Z., Scavino, M., and Tempone, R. (2016). Supplementary material for “A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions” Bayesian Analysis.
  • Samarskii, A. A. and Vabishchevich, P. N. (2007). Numerical Methods for Solving Inverse problems of Mathematical Physics. Walter de Gruyter.
  • Wang, J. and Zabaras, N. (2004). “A Bayesian inference approach to the inverse heat conduction problem.” International Journal of Heat and Mass Transfer, 47(17–18): 3927–3941.
  • Wang, J. and Zabaras, N. (2005). “Hierarchical Bayesian models for inverse problems in heat conduction.” Inverse Problems, 21(1): 183–206.
  • Xun, X., Cao, J., Mallick, B., Maity, A., and Carroll, R. J. (2013). “Parameter estimation of partial differential equation models.” Journal of the American Statistical Association, 108(503): 1009–1020.

Supplemental materials

  • Supplementary material for “A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions”. Supplementary material includes proofs of Theorem 7, Theorem 8 and Remark 14.